Real function/Increasing and decreasing/Definition

Increasing function

Let $I\subseteq ℝ$ denote an interval and let

$f:I⟶ℝ$

denote a function. Then $f$ is called increasing if

$f(x^{\prime })\ge f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }\ge x,$

strictly increasing if

$f(x^{\prime })>f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }>x,$

decreasing if

$f(x^{\prime })\le f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }\ge x,$

strictly decreasing if

$f(x^{\prime })<f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }>x.$